We extend the discrete Regge action of causal dynamical triangulations to include discrete versions of the curvature squared terms appearing in the continuum action of (2+1)-dimensional projectable Hořava-Lifshitz gravity. Focusing on an ensemble of spacetimes whose spacelike hypersurfaces are 2-spheres, we employ Markov chain Monte Carlo simulations to study the path integral defined by this extended discrete action. We demonstrate the existence of known and novel macroscopic phases of spacetime geometry, and we present preliminary evidence for the consistency of these phases with solutions to the equations of motion of classical Hořava-Lifshitz gravity. Apparently, the phase diagram contains a phase transition between a time-dependent de Sitter-like phase and a time-independent phase. We speculate that this phase transition may be understood in terms of deconfinement of the global gravitational Hamiltonian integrated over a spatial 2-sphere.

We study a lattice regularization of the gravitational path integral-causal dynamical triangulationsfor (2 + 1)-dimensional Einstein gravity with positive cosmological constant in the presence of past and future spacelike boundaries of fixed intrinsic geometries. For spatial topology of a 2-sphere, we determine the form of the Einstein-Hilbert action supplemented by the Gibbons-Hawking-York boundary terms within the Regge calculus of causal triangulations. Employing this action we numerically simulate a variety of transition amplitudes from the past boundary to the future boundary. To the extent that we have so far investigated them, these transition amplitudes appear consistent with the gravitational effective action previously found to characterize the ground state of quantum spacetime geometry within the Euclidean de Sitter-like phase. Certain of these transition amplitudes convincingly demonstrate that the so-called stalks present in this phase are numerical artifacts of the lattice regularization, seemingly indicate that the quantization technique of causal dynamical triangulations differs in detail from that of the no-boundary proposal of Hartle and Hawking, and possibly represent the first numerical simulations of portions of temporally unbounded quantum spacetime geometry within the causal dynamical triangulations approach. We also uncover tantalizing evidence suggesting that Lorentzian not Euclidean de Sitter spacetime dominates the ground state on sufficiently large scales.

The entanglement entropy of quantum fields across a spatial boundary is UV divergent, its leading contribution proportional to the area of this boundary. We demonstrate that the Callan-Wilczek formula provides a renormalized geometrical definition of this entanglement entropy for a class of quantum states defined by a path integral over quantum fields propagating on a curved background spacetime. In particular, UV divergences localized on the spatial boundary do not contribute to the entanglement entropy, the leading contribution to the renormalized entanglement entropy is given by the Bekenstein-Hawking formula, and subleading UV-sensitive contributions are given in terms of renormalized couplings of the gravitational effective action. These results hold even if the UV-divergent contribution to the entanglement entropy is negative, for example, in theories with non-minimal scalar couplings to gravity. We show that subleading UV-sensitive contributions to the renormalized entanglement entropy depend nontrivially on the quantum state. We compute new subleading UV-sensitive contributions to the renormalized entanglement entropy, finding agreement with the Wald entropy formula in all cases. We speculate that the entanglement entropy of an arbitrary spatial boundary may be a welldefined observable in quantum gravity.

The causal dynamical triangulations approach aims to construct a quantum theory of gravity as the continuum limit of a lattice-regularized model of dynamical geometry. A renormalization group schemein concert with finite size scaling analysis-is essential to this aim. Formulating and implementing such a scheme in the present context raises novel and notable conceptual and technical problems. I explored these problems, and, building on standard techniques, suggested potential solutions in the first paper of this two-part series. As an application of these solutions, I now propose a renormalization group scheme for causal dynamical triangulations. This scheme differs significantly from that studied recently by Ambjørn, Görlich, Jurkiewicz, Kreienbuehl, and Loll.

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